Spectral analysis of integrodierential equations

arising in viscoelasticity

N. A. Rautian, V. V. Vlasov

(Lomonosov Moscow State University)

OTIND-2016, December 1720, 2016

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Introduction

We study integro-dierential equations with unbounded operator coecientsin Hilbert space. The main part of these equations is an abstract hyperbolicequations, disturbed by the terms containing abstract integral Volterraoperators. The equations mentioned above are the abstract form of theintegro-dierential equation of Gurtin-Pipkin (see bibliography cited belowfor more details) describing the process of heat propagation in media withmemory, process of wave propagation in the visco-elastic media, and alsoarising in the problems of porous media (Darci law).We obtain correct solvability of the initial value problems for the describedequations in the weighted Sobolev spaces on the positive semiaxis.We analyse spectral problems for the operatorvalued functions which are thesymbols of these equations. Moreover we study the spectrum of the abstractintegro-dierential equation of Gurtin-Pipkin type.

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Let us H be a separable Hilbert space and A be a self-adjoint positiveoperator A∗ = A > κ0 (κ0 > 0) acting in the spaceH and having a compactinverse operator. Let us B be a symmetric operator (Bx, y) = (x,By),acting in the space H having the domain Dom (B) (Dom (A) ⊆ Dom (B)).Moreover B be a nonnegative operator that is (Bx, x) > 0 for any x, y ∈Dom (B) and satisfying to inequality ‖Bx‖ 6 κ ‖Ax‖, 0 < κ < 1 for anyx ∈ Dom (A) and I be the identity operator acting in the space H.We consider the following problem for a second-order integrodierentialequation on the semiaxis R+ = (0,∞):

d2u(t)

dt2+Au(t)+Bu(t)−

∫ t

0K(t− s)Au(s)ds−

∫ t

0Q(t− s)Bu(s)ds =

= f(t), t ∈ R+, (1)

u(+0) = ϕ0, u(1)(+0) = ϕ1. (2)

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Assume that the scalar functions K(t) and Q(t) that are the kernels ofintegral operators admits the following representations:

K(t) =

∞∑k=1

ake−γkt, Q(t) =

∞∑k=1

bke−γkt, (3)

where ak > 0, bk > 0, γk+1 > γk > 0, k ∈ N, γk → +∞ (k → +∞). Weassume that the following conditions are true:

∞∑k=1

akγk

< 1,

∞∑k=1

bkγk

< 1. (4)

The conditions (4) means that K(t), Q(t) ∈ L1(R+), ‖K‖L1< 1, ‖Q‖L1

<1. If conditions (4) are supplemented with the conditions

K(0) =

∞∑k=1

ak < +∞, Q(0) =

∞∑k=1

bk < +∞. (5)

then the kernels K(t) and Q(t) belong to the space W 11 (R+).

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The equation (1) can be regarded as an abstract form of dinamicalviscoelastic integrodierential equation where operatoprs A and B aregenerated by the following dierential expressions

A = −ρ−1µ

(∆u+

1

3grad(divu)

), B = −1

3ρ−1λ · grad(divu),

here u = ~u(x, t) ∈ R3 is displacement vector of viscoelastic hereditaryisotropic media that ll the bounded domain Ω ⊂ R3 with smooth boundary,∂Ω, ρ is a constant density, ρ > 0, Lame parameters λ, µ are the positiveconstants, K(t), Q(t) are the relaxation functions characterizing hereditaryproperties of media. On the domain boundary ∂Ω the Dirichlet condition

u|∂Ω = 0. (6)

is satised. The Hilbert space H can be realized as the space of threedimensional vector-functions L2(Ω). The domain Dom(A) belongs to theSobolev space W 2

2 (Ω) of vector functions satisfying the condition (6).

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In case operator B = 0, positive and self-adjoint operator A can be realizedas operator Ay = −y′′(x), where x ∈ (0, π), y(0) = y(π) = 0, or theoperator Ay = −∆y with Dirichlet conditions on the bounded domain Q ⊂Rn with smooth boundary (H = L2(Q)) or more general elliptic self-adjointoperators in the space L2(Q). The equation (1) can be regarded as anabstract form of the Gurtin-Pipkin equation that describes heat transfer inmaterials with memory with nite speed.

1) Gurtin M. E., Pipkin A. C. General theory of heat conduction with nitewave speed // Arch. Rat. Mech. Anal., 1968, V. 31, P. 113126.

2) Pruss J. Evolutionary Integral Equations and Applications// Monographsin Mathematics, 1993, V.87, Birkhauser Verlag. Basel-Baston-Berlin.

3) Amendola G., Fabrizio M., Golden, J. M. Thermodynamics of materialswith memory: theory and applications. New York: Springer, 2012.

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Applying the Laplace transform to the equation (1) with zero initialconditions we obtain the following operator-valued function

L(λ) = λ2I +A+B − K(λ)A− Q(λ)B, (7)

which are the symbol (analogue of the characteristic quasi-polynomial) ofthe equation (1). Here K(λ) and Q(λ) are the Laplace transforms of kernelsK(t) and Q(t) respectively, having the following representations

K(λ) =

∞∑k=1

ak(λ+ γk)

, Q(λ) =

∞∑k=1

bk(λ+ γk)

, (8)

Denition

The set of values λ ∈ C is called the resolvent set R(L) of operator-valuedfunction L(λ) if there exists L−1(λ) is bounded for any λ ∈ R(L). The setσ(L) = λ ∈ C\R(L) |L(λ) exists is called the spectra of operator-valuedfucntion L(λ).

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Denote byA0 := A+B. It is follows from the properties of operatorsA andBthat the operator A0 is positive and self-adjoint. Moreover A0 is reversible,operators AA−1

0 , BA−10 are bounded and operator A−1

0 is compact (seemonograph T. Kato Perturbation Theory for Linear Operators// Springer-Verlag Berlin Heidelberg New York, 1980).Let us denote by Wn

2,γ (R+, A0) the Sobolev space of the vector-valuedfunctions on the positive semiaxis R+ = (0,∞) with the values in the spaceH equiped by the norm

‖u‖Wn2,γ(R+,A0) ≡

(∫ ∞0

e−2γt

(∥∥∥u(n)(t)∥∥∥2

H+ ‖A0u(t)‖2H

)dt

)1/2

,

γ ≥ 0.

For more detail description of the space Wn2,γ (R+, A0) see the monograph

J. L. Lions and E. Magenes Nonhomogeneous Boundary-Value Problems and

Applications // Springer-Verlag, Berlin-Heidelberg-New York. 1972, chapter1. For n = 0 we haveW 0

2,γ (R+, A0) ≡ L2,γ (R+, H), and for γ = 0 we shallwrite Wn

2,0 = Wn2 .

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Correct solvability

We establish well-dened solvability of initial boundary value problem (1),(2) in weighted Sobolev spaces on the positive semi-axis and examine thespectra localization of operator-valued functions L(λ) representing symbolof the equation (1).

Denition

Vector-valued function u is called the strong solution of the problem (1),(2), if it belongs to the space W 2

2,γ(R+, A0) for some γ > 0, satises theequation (1) almost everywhere on the semiaxis R+, and also initialconditions (2).

Let us convert the domain Dom(Aβ0 ) of the operator Aβ0 , (β > 0) into the

Hilbert space Hβ , by introducing the norm ‖ · ‖β = ‖Aβ0 · ‖ on the space

Dom(Aβ0 ) which is equivalent the graph norm of the operator Aβ0 .The following theorem present the result on the correct solvability of theproblem (1), (2).

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Theorem

Suppose that f (1)(t) ∈ L2,γ0 (R+, H) for some γ0 > 0 and the condition

(4) is satised, moreover ϕ0 ∈ H1, ϕ1 ∈ H1/2. Then there exists such

γ1 ≥ γ0 that the problem (1), (2) has the unique solution in the space

W 22,γ (R+, A0) for arbitrary γ > γ1. Moreover the following estimate is valid

‖u‖W 22,γ(R+,A0) ≤ d

(∥∥∥f (1)(t)∥∥∥L2,γ(R+,H)

+ ‖A0ϕ0‖H +∥∥∥A1/2

0 ϕ1

∥∥∥H

)(9)

with a constant d that does not depend on vector-function f and vectors

ϕ0 and ϕ1.

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Spectral Analysis

Let us study the structure and localization of the spectra of operator-valuedfunction L(λ) when the conditions (4) and (5) are satised. We shall supposethat the following assumptions on the sequence γk∞k=1 are valid:

supk∈N

γ2k(γk+1 − γk) = +∞, (10)

limk→∞

γk − γk−1

γk= 0. (11)

Remark that condition (11) is satised when the sequence γk ' kα, α > 0.

Really in this caseγk − γk−1

γk∼ α

k→ 0, (k →∞). These power asymptotics

arising in averaging theory where terms of the sequence γk∞k=1 are pointsof the spectrum of a special elliptic Stokes-type problem with periodicconditions (see the monograph E. Sances Palensia Nonhomogeneous Media

and Oscillation Theory Springer-Verlag Berlin Heidelberg New York, 1980). Inturn the condition (11) is not satised if the sequence γk = cqk, q > 1, c > 0.Such behavior of the sequence γk∞k=1 is rather seldom in applications.

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Let us determine the localization of nonreal roots of the equation

(L(λ)f, f) = λ2 + (Af, f) + (Bf, f)−∞∑k=1

ak(Af, f) + bk(Bf, f)

λ+ γk= 0,

(f ∈ D(A), ||f || = 1) (12)

Let us introduce the following notations:ω2 = ((A + B)f, f) > 0, ck = ((akA+ bkB)f, f)ω−2 > 0, A(λ) =∞∑k=1

ak(λ+ γk)−1, B(λ) =

∞∑k=1

bk(λ+ γk)−1.

Under these notations the equation (12) take a form:

λ2

ω2+ 1 =

∞∑k=1

ckλ+ γk

, λ ∈ C. (13)

It is easy show that the conditions∞∑k=1

ck < +∞,∞∑k=1

ckγ−1k < 1 are valid.

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Then we use the following lemma:

Lemma (1)

Suppose the conditions (10), (11) are satised and the series∞∑k=1

ck is

convergent. Then the equation (13) has two complex conjugate roots

λ±0 = α0 ± iβ0 ∈ C, α0, β0 ∈ R, α0 < 0 and the following inequalites

− 1

2

∞∑k=1

ck 6 α0 6 −1

2

∞∑k=1

ω2ckω2 + γ2

k

. (14)

are valid for real part α0 of roots λ±0 .

In our previous notations the inequalities (14) have the following form

−1

2

∞∑k=1

((akA+ bkB)f, f)

((A+B)f, f)6 α0 6 −1

2

∞∑k=1

((akA+ bkB)f, f)((A+B + γ2

kI)f, f) ,

where f ∈ D(A), ||f || = 1 and α0 is real part of nonreal zeroes of theequation (12).

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Let us determine the location of real zeroes of the equation (12). Denoteby τ = (Af, f)ω−2, 0 6 inf τ 6 τ 6 sup τ 6 1. Rewrite the (12) in thefollowing form

λ2

ω2+1 = τ

∞∑k=1

akλ+ γk

+(1−τ)

∞∑k=1

bkλ+ γk

=: τA(λ)+(1−τ)B(λ). (15)

Consider the eqiation

Φτ (p) := τA(p) + (1− τ)B(p) = 1, (16)

The function Φτ (p) → ∞ for p → −γk, k ∈ N and it is monotonouslydecreases for real p ∈ (−γk,−γk−1), k ∈ N (γ0 = 0), hence the equation(16) has the innity consequence of real zeroes pk(τ) ∈ (−γk,−γk−1),k ∈ N. In turn the equation (15) also has the innity consequence ofreal zeroes λk(τ) ∈ (−γk, pk(τ)) (by constraction), therefore λk(τ) ∈(−γk, max

0<τ<1pk(τ)) for any τ ∈ (0, 1), k ∈ N.

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Lemma (2)

Suppose the conditions (10), (11) are satised. Then real zeroes of the

equation (12) belong to intervals ∆k = (−γk, pk) where

pk = max pk(τ ′), pk(τ ′′), pk(τ) are real zeroes of the equation (16)belonging to intervals (−γk,−γk−1), k ∈ N (γ0 = 0),

τ ′ :=∥∥A−1/2A0A

−1/2∥∥−1

, τ ′′ :=∥∥∥A−1/2

0 AA−1/20

∥∥∥.Remark. According to lemma 2.1 from the article A.A. Shkalikov Strongly

Damped Pencils of Operators and Solvability of the Corresponding Operator-

Dierential Equations (Mathematics of the USSR-Sbornik (1989), 63(1):97)operator A−1/2BA−1/2 admits bounded closure in the space H. Hence weobtain that operator A−1/2A0A

−1/2 = I + A−1/2BA−1/2 admits boundedclosure in the space H. In turn owing to lemma 2.1 from cited article and

due to selfadjointness of operator A0 = A+B operator A−1/20 AA

−1/20 also

admits bounded closure in the space H. Thus values τ ′ and τ ′′ are denedcorrectly.

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Theorem (The main spectral theorem)

Suppose that the conditions (4), (5), (10), (11) are satised. Then the

spectra of operator-valued function L(λ) belongs to the unit intervals

∆k = (−γk, pk] ⊂ (−γk,−γk−1), k ∈ N (γ0 = 0) and the strip

λ ∈ C|α1 6 Reλ 6 α2, where pk = max pk(τ ′), pk(τ ′′), pk(τ) are realzeroes of the equation

Φτ (p) := τ∞∑k=1

ak(p+ γk)−1+(1−τ)

∞∑k=1

bk(p+ γk)−1 = 1, (0 ≤ τ ≤ 1).

belonging the intervals (−γk,−γk−1), k ∈ N (γ0 = 0),

τ ′ :=∥∥A−1/2A0A

−1/2∥∥−1

, τ ′′ :=∥∥∥A−1/2

0 AA−1/20

∥∥∥, (0 < τ ′ < τ ′′ 6 1),

α1 = −1

2sup‖f‖=1

∞∑k=1

((akA+ bkB)f, f)

((A+B)f, f), f ∈ D(A),

α2 = −1

2inf‖f‖=1

∞∑k=1

((akA+ bkB)f, f)((A+B + γ2

kI)f, f) , f ∈ D(A).

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Remark

Constants α1 and α2 from the main spectral theorem satisfy the following

estimates

α1 > −1

2

∥∥∥∥∥A−1/20

( ∞∑k=1

akA+

∞∑k=1

bkB

)A−1/20

∥∥∥∥∥ ,α2 < −

1

2

∥∥∥(a1A+ b1B)−1/2 (A0 + γ21I)

(a1A+ b1B)−1/2∥∥∥−1

.

Theorem

Unreal part of the spectra of operator-valued function L(λ) is symmetric

concerning the real axis and consists of eigenvalues of nite algebraic

multipliciy. Moreover for any ε > 0 in the domain

Ωε := C\ λ : α1 ≤ Reλ ≤ α2, | Imλ| < ε, eigenvalues are isolated that

is have no accumulation points.

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Generalizations

We consider the following problem (1), (2)

d2u(t)

dt2+Au(t)+Bu(t)−

∫ t

0K(t− s)Au(s)ds−

∫ t

0Q(t− s)Bu(s)ds =

= f(t), t ∈ R+,

u(+0) = ϕ0, u(1)(+0) = ϕ1,

in assumption that the scalar functions K(t) and Q(t) admit the followingrepresentations:

K(t) =

∫ ∞0

e−tτdµ(τ), Q(t) =

∫ ∞0

e−tτdη(τ), (17)

where dµ and dη are the positive measures corresponding to an increasingright-continuous distribution functions µ and η respectively. The integral isunderstood in the Stieltjes sense.

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We assume that the following conditions are true:

0 <

∫ ∞0

dµ(τ)

τ< 1, 0 <

∫ ∞0

dη(τ)

τ< 1, (18)

K(0) =

∫ ∞0

dµ(τ) ≡ Varµ|∞0 < +∞,

Q(0) =

∫ ∞0

dη(τ) ≡ Var η|∞0 < +∞. (19)

Here the supports µ and η belong to the interval (d0,+∞), d0 > 0.

Lemma

Suppuse that conditions (18), (19) holds. Then the operator-function L(λ)is invertible in closed right half-plane and the following estimate∥∥∥A1/2L−1(λ)A1/2

∥∥∥ 6 const, Reλ > γ > 0.

is valid.

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We formulate the results about the spectrum localization of operator-function L(λ) when the measures dµ(τ), dη(τ) have compact supports.

Theorem

Suppuse that conditions (18), (19) holds and the supports of measures

dµ(τ), dη(τ) belong to the segment [d1, d2], 0 < d1 < d2 < +∞. Then for

arbitrary θ0 > 0 there exist number R0 > 0, such that spectrum of

operator-function L(λ) belongs to the set

Ω = λ ∈ C : Reλ < 0, |λ| < R0 ∪ λ ∈ C : α1 ≤ Reλ ≤ α2 ,

where α1 = α0 − θ0, R0 > max(d2,−α0 + θ0),

α0 = −1

2sup‖f‖=1

∞∑k=1

((K(0)A+Q(0)B)f, f)

((A+B)f, f), f ∈ D(A),

α2 = −1

2inf‖f‖=1

∞∑k=1

((K(0)A+Q(0)B)f, f)((A+B + d2

2I)f, f) , f ∈ D(A). (20)

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Remark

Let us conditions of the previous theorem hold. Then there exists such

γ0 > 0, that operator-function L−1(λ) satises the following estimate∥∥L−1(λ)∥∥ 6

const

|λ||Reλ|(21)

on the set λ : Reλ < −R0 ∪ λ : Reλ > γ0.

Remark

The quantity α0 in the statement of previous theorem can be estimated as

α0 > −1

2

∥∥∥A−1/20 (K(0)A+Q(0)B)A

−1/20

∥∥∥ .

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Theorem

Let us conditions of the previous theorem hold. Then the nonreal spectrum

of the operator-function L(λ) is symmetric with respect to the real axis

and consist of eigenvalues of nite algebraic multiplicity, moreover for any

ε > 0 in the domain

Ωε := Ω\ λ ∈ C : −d2 − ε < Reλ < 0, | Imλ| < ε

eigenvalues is isolated i.e., have no points of accumulation.

These results was proved in the articles1) V.V. Vlasov and N.A. Rautian, Correct solvability and spectral analysis ofintegrodierential equations arising in the viscoelasticity theory // Sovrem.mat. Fundam. napravl. [Contemp. Math. Fundam. Directions], 2015, 58,2242. (in Russian)2) V.V. Vlasov and N.A. Rautian, Spectral Analysis of IntegrodierentialEquations in a Hilbert Space // Sovrem. mat. Fundam. napravl. [Contemp.Math. Fundam. Directions], 2016, 62, 5371. (in Russian)

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In our prevoius works1) V. V. Vlasov, N. A. Rautian Well-Dened Solvability and Spectral

Analysis of Abstract Hyperbolic Integrodierential Equations // Journal ofMathematical Sciences, 179:3 (2011), P. 390414,2) V. V. Vlasov, N. A. Rautian, A. S. Shamaev Spectral analysis and correct

solvability of abstract integrodierential equations arising in thermophysics

and acoustics // Journal of Mathematical Sciences, 190:1 (2013), P. 3465,3) V. V. Vlasov, N. A. Rautian Spectral Analysis and Representations of

Solutions of Abstract Integro-dierential Equations in Hilbert Space //Operator Theory: Advances and Applications. Springer Basel AG, V.235,2013, P. 519537,we considered in detail the case when B = 0. In this case the equation(1) has the abstract form of Gurtin-Pipkin integro-dierential equation thatdescribe heat transfer in materials with memory with nite speed and has anumber of other applications.

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Previous results

In our previous works we considered in detail the case when B = 0. Inthis case the equation (1) has the abstract form of Gurtin-Pipkin integro-dierential equation that describe heat transfer in materials with memorywith nite speed and has a number of other applications. Then the problem(1), (2) has the following form

d2u(t)

dt2+A2u(t)−

∫ t

0K(t− s)A2u(s)ds = f(t), t ∈ R+,

u(+0) = ϕ0, u(1)(+0) = ϕ1.

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Let us denote by en∞n=1 the orthonormal basis consisting of eigenvectorsof operator A2 corresponding to the eigenvalues aj : A

2en = a2nen, n ∈ N.

The eigenvalues a2n are numerated in increasing order 0 < a2

1 < a22 < ...;

a2n → +∞ for n→ +∞.

Consider the projection ln(λ) := (L(λ)en, en) = λ2 + a2n

(1− K(λ)

)of

the operator-valued function L(λ) on the one-dimensional subspace formedby the vector en. Thus we obtain the countable set of the meromorphicfunctions ln(λ), n ∈ N. Then the spectrum of the operator-valued functionL(λ) is the closure of the zeroes set of the functions ln(λ)∞n=1.

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Let us denote by en∞n=1 the orthonormal basis consisting of eigenvectorsof operator A2 corresponding to the eigenvalues aj : A

2en = a2nen, n ∈ N.

The eigenvalues a2n are numerated in increasing order 0 < a2

1 < a22 < ...;

a2n → +∞ for n→ +∞.

Consider the projection ln(λ) := (L(λ)en, en) = λ2 + a2n

(1− K(λ)

)of

the operator-valued function L(λ) on the one-dimensional subspace formedby the vector en. Thus we obtain the countable set of the meromorphicfunctions ln(λ), n ∈ N. Then the spectrum of the operator-valued functionL(λ) is the closure of the zeroes set of the functions ln(λ)∞n=1.

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Theorem

Let us suppose B = 0 and conditions (4), (10) are satised. Then the

spectrum of the operator-valued function L(λ) is the closure of the zeroes

set of the functions ln(λ)∞n=1 that is

σ(L) := λk,n ∈ R|k ∈ N, n ∈ N ∪λ±n |n ∈ N

, (22)

where λ±n are nonreal conjugate complex zeroes λ+n = λ−n of functions

ln(λ). Moreover the real zeroes satisfy the inequalities

...− γk+1 < xk+1 < λk+1,n < −γk < ... < −γ1 < x1 < λ1,n < 0, k ∈ N,(23)

where xk are the real zeroes of the function 1− K(λ) and

λk,n = xk +O(1/a2

n

). Moreover if the condition (5) is satised then the

conjugate complex zeroes λ±n are asymptotically represented in the form

λ±n = ±i(an +O

(1

an

))− 1

2

∞∑k=1

ak +O

(1

a2n

), an → +∞. (24)

27 / 42

The following picture represents the spectral structure of the operator-function L(λ) in the complex plane λ, where β = −K(0)/2.

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The following theorem presents the asymptotic of the complex zeroes of thefunction ln(λ) when an → +∞, in the case B = 0 when the condition (5)is not satised, and the sequences ck∞k=1 and γk∞k=1 have the followingasymptotic representation

ck =A

kα+O

(1

kα+1

), (25)

γk = Bkβ +O(kβ−1

),

for k → +∞, where ck > 0, γk+1 > γk > 0, k ∈ N and the constantsA > 0, B > 0, 0 < α 6 1, α+ β > 1.

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Let us denote the following constant by

D := − i2

∫ ∞0

dt

tr(i+ t)=

π

2 sin(πr)· eiπ

2(1− r)

,

where 0 < r < 1.

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Theorem

Let us suppose B = 0 and conditions (25) are satised. Then conjugate

complex zeroes λ±n , λ+n = λ−n of the function ln(λ) are asymptotically

represented in the following form

λ±n = ±ian −DA

βB1−r a1−rn +O(a1−2r

n ), for 0 < r <1

2

λ±n = ±ian −DA

βB1−r a1−rn +O(1), for

1

26 r < 1,

λ±n = ±ian −1

2

A

βln an +O (1) , for r = 1,

for n→ +∞, where r :=α+ β − 1

β, the constant D depends on r.

The following picture represents the spectral structure of the operator valuedfunction L(λ) in the complex plane λ in case B = 0 and K(t) ∈ L1(R+),K(t) /∈W 1

1 (R+).

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Imλ

Reλ

λ±n

x2x3 0

−∞

Ðèñ.: Spectral structure in case B = 0, K(t) ∈ L1(R+), K(t) /∈W 11 (R+).

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Theorem

Let us suppose B = 0, the conditions (4), (10) are satised, but thecondition (5) is not satised. Then the pare of the conjugate complex

zeroes λ±n , λ+n = λ−n of the meromorphic function ln(λ) asymptotically

represented in the form

λ±n = ±iΘ · an + Φ(an, ck∞k=1, γk∞k=1), k ∈ N (26)

where Θ = Θ(ck∞k=1, γk∞k=1) is a positive constant, depending on the

sequences ck∞k=1, γk∞k=1, Re Φ = O(an), Im Φ = o(an) while

an → +∞ and liman→∞

Re Φ(an, ck∞k=1, γk∞k=1) = −∞.

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Representation of the solutions

On the base of the spectral theorems we obtain the representation of thesolution of the problem (1), (2) when operator B = 0 in the form of theseries.

Theorem

Let us suppose that f(t) = 0 for t ∈ R+, vector-function

u(t) ∈W 22,γ (R+, A), γ > 0 is a strong solution of the problem (1), (2) and

the conditions (4), (10) are satised. Then, for arbitrary t ∈ R+ the

solution u(t) of the problem (1), (2) is represented in the following series

u(t) =

∞∑n=1

(ωn(t, λ+

n ) + ωn(t, λ−n ) +

∞∑k=1

ωn(t, λkn)

)en, (27)

that is convergent by the norm of the space H, where

ωn(t, λ) =(ϕ1n + λϕ0n) eλt

l(1)n (λ)

,

ϕ0n = (ϕ0, en), ϕ1n = (ϕ1, en).34 / 42

Theorem

Let us suppose vector-function f(t) ∈ C ([0, T ], H) for arbitrary T > 0,vector-function u(t) ∈W 2

2,γ

(R+, A

2), γ > 0 is a strong solution of the

problem (1), (2) and the conditions (4), (10), ϕ0 = ϕ1 = 0 are satised.

Then, for arbitrary t ∈ R+ the solution u(t) of the problem (1), (2) isrepresented in the following series

u(t) =

∞∑n=1

(ωn(t, λ+

n ) + ωn(t, λ−n ) +

∞∑k=1

ωn(t, λkn)

)en, (28)

that is convergent by the norm of the space H, where

ωn(t, λ) =

t∫0

fn(τ)eλ(t−τ)dτ

l(1)n (λ)

.

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Examples of very unstable linear partial functional

dierential equations

We consider on the positive semiaxis R+ = (0,∞) the following initialproblem for the integro-dierential equation of the rst order

dv

dt+

t∫−∞

A2v(t− s)dσ(s) = f(t), t ∈ R+, (29)

v(t) = 0, t ∈ (−∞, 0), (30)

where dσ is a positive measure. We identify this measure with its distributionfunction σ(s), so σ(s) is increasing, continuous from the right, and theintegral is interpreted as a Stieltjes integral.

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There is an important particular case when the distribution function σ(t) canbe represented in the following form σ(t) = θ(t − h), where h > 0. In thismodel case the integral term in the equation (29) has the following form

t∫−∞

A2u(t− s)dσ(s) = A2u(t− h).

Thus equation (29) is the delay equation. The operator valued functionL(λ) = λI +A2e−λh is the symbol of this equation.

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In a case when H space has the nite dimension, the considered delayequation obtained was investigated by many authors. However, in a casewhen H has innite dimension and A is the unlimited self-conjugate positiveoperator in Hilbert space of H, the spectum of the operator-function L(λ),apparently, wasn't studied earlier.

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Let us consider the following model example. Let's assume that opertor A2

is realized as follows: A2y(x) = −y(2)xx (x), y(0) = y(π) = 0, H = L2(0, π)

and h = 1. In this case an = n, and spectrum of operator-function L(λ) isthe closure of the zeroes set of the functions ln(λ) = λ + n2e−λ, n ∈ N,that is

σ(L) =⋃n∈N

⋃k∈Z

λnk, ln(λnk) = 0.

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The asymptotic representation of zeroes λnk for k → +∞ and xed nis well-known, however it is interesting and unexpected that there existsthe consequence of eigenvalues of λnk(n), such that Reλnk(n) → +∞ forn→ +∞. Indeed, let us x n and extract the real and imaginary part fromexpression λeλ, λ = x+ iy. Then the equation λeλ = −n2 is equivalent tothe following system:

ex(x cos y − y sin y) = −n2,

x sin y + y cos y = 0.(31)

The analysis of system (31) shows that it has solutions, asymptoticallyrepresentable as follows

xn ≈ 2 lnn−ln lnn→ +∞, yn ≈ π(

1− 1

2 lnn− ln lnn

), n→ +∞.

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Thus, the problem (29), (30) has the solutions of the following typee(xn+iyn)ten, increasing for t → +∞ quicker than any xed exponent eγt,where γ > 0 is a constant. So the problem (29), (30) is unstable andit isn't correctly solvable in Sobolev space Wm

2,γ(R+, A2) for any γ > 0

м m ∈ N. Let's notice that for h = 0 we have the classic mixedproblem for heat equation that is stable and correctly solvable in the Sobolevspace W 1

2,0(R+, A2). It should be noted also that the system (31) was

investigated earlier and it was established that the system (31) has suchsolution λnk(n) = xn + iyn that xn → 0, n→ +∞.

Examples of very unstable linear partial functional dierential equations aredescribed in the work

R. S. Ismagilov, N. A. Rautian, V. V. Vlasov, Examples of very unstable linearpartial functional dierential equations // http://arxiv.org/abs/1402.4107

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Thank you very much for your attention.

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